Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C92×C37 · Schur multiplier (H2) = C3×C9 |
| direct product | C37×#10624470 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C37 | C9⋊C9 |
| commutator | C9 | C92×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| socle | C3×C37 | #6855 |
| prime radical | C33×C37 | H(3,𝔽3) |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C37 ◁ C32×C9×C37 ◁ G | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 118 points:
(1,2,6)(3,10,30,13,33,55,14,34,64)(4,15,43)(5,9,22)(7,23,53,25,56,39,26,57,72)(8,27,48)(11,35,68,37,69,79,42,74,54)(12,38,50,40,32,51,41,66,19)(16,46,61)(17,45,20)(18,29,21)(24,58,71,59,63,73,60,80,36)(28,62,47)(31,49,70,65,76,78,67,77,81)(44,52,75)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(1,3,11)(2,7,24,9,26,59,29,25,60)(4,12,36)(5,14,37)(6,19,49,21,50,77,22,51,76)(8,10,31,15,34,65,45,33,67)(13,42,18)(16,39,70)(17,41,71)(20,23,54,43,56,79,27,57,68)(28,32,35,44,38,69,46,66,74)(30,63,52,55,58,62,64,80,61)(40,73,48)(47,53,78)(72,81,75)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 30);